# PDE Seminar

## Blow up solutions for a mean field equation on sphere and torus

### 胡烨耀 博士 ()

#### 2018年6月26日下午2:00-3:00 闵行校区行政楼12楼PDE中心1202报告厅

Abstract: We consider the mean field equation

$$ \Delta_g u + \rho\left(\frac{h e^u}{\int_M h e^u}-\frac{1}{|M|}\right)=0, $$ where $h$ is a positive function, $(M,g)$ is a closed Riemann surface, $\Delta_g$ is the associated Laplace-Beltrami operator and $|M|$ is the total area of the surface. It is well known that non-degeneracy condition on the $m$-vertex Hamiltonian $f_h$ involving $h$, the curvature and the Green function implies the existence of blow-up solutions as $\rho\rightarrow 8\pi m$ where $m$ is any positive integer. We construct blow-up solutions of the degenerate cases when the underlying manifold $M$ is the sphere or torus and $h\equiv 1$ by assuming some additional symmetry. This is a joint work with Ze Cheng (UTSA) and Changfeng Gui (UTSA).